36
3 Effects of the Introduction of Superconductivity into Electromagnetism
B x = −
∂A y
∂z
=
μ 0 aI
−x
2
+ z
2
− a
2
π
x 2 + (z − a)
2
x 2 + (z + a)
2
,
B y =
∂A x
∂z
−
∂A z
∂x
= 0,
B z =
∂A y
∂x
= −
2μ 0 aIxz
π
x 2 + (z − a)
2
x 2 + (z + a)
2
.
(3.19)
In particular, the magnetic flux density on the superconductor surface (z = 0) is
B x = −
μ 0 aI
π
x 2 + a 2
, B y = B z = 0.
(3.20)
Thus, it is shown shows that the magnetic flux density is parallel to the surface. Using
(3.8), the current density on the superconductor surface is obtained as
τ =
1
μ 0
B x (z = 0) = −
aI
π
x 2 + a 2
.
(3.21)
The total current on the surface is
−
∞
−∞
aI dx
π
x 2 + a 2
= −
I
π
π/2
−π/2
dθ = −I ,
(3.22)
which is equal to the image current. Thus, this phenomenon is similar to the electric
phenomenon of the conductor. The magnetic flux density outside the superconductor
is schematically shown in Fig. 3.7. The difference from the electric field in Fig. 3.6
comes from the different nature of the field.
Fig. 3.7 Magnetic flux
density outside the
superconductor